Numbers of solutions of congruences: Poincaré series for strongly nondegenerate forms

Abstract:

Let $p$ be a fixed prime, $f({x_1}, \ldots ,{x_k})$ a polynomial over ${{\mathbf {Z}}_p}$, the $p$-adic integers. ${c_n}$ the number of solutions of $f = 0$ over ${\mathbf {Z}}/{p^n}{\mathbf {Z}}$ and ${P_f}(t) = \sum \nolimits _{i = 0}^\infty {{c_i}{t^i}}$ the Poincaré series. A general approach to computing ${c_n}$ and ${P_f}(t)$ is given and explicit formulas are derived for strongly nondegenerate forms.